27–30. Predator-prey models Consider the following pairs of differential equations that model a predator-prey system with populations x and y. In each case, carry out the following steps.
c. Find the equilibrium points for the system.


x′(t) = −3x + xy, y′(t) = 2y − xy

Verified step by step guidance

1. Write down the system of differential equations clearly: \(x'(t) = -3x + xy\) and \(y'(t) = 2y - xy\).
2. To find equilibrium points, set both derivatives equal to zero because equilibrium occurs where the populations do not change: \(x'(t) = 0\) and \(y'(t) = 0\).
3. From \(x'(t) = 0\), we have \(-3x + xy = 0\). Factor this expression to get \(x(-3 + y) = 0\).
4. From \(y'(t) = 0\), we have \$2y - xy = 0\(. Factor this expression to get \)y(2 - x) = 0$.
5. Solve the system of equations \(x(-3 + y) = 0\) and \(y(2 - x) = 0\) by considering cases where each factor is zero to find all equilibrium points \((x, y)\).

Key Concepts

Equilibrium Points in Differential Equations

Equilibrium points occur where the rates of change of all variables are zero, meaning the system is in a steady state. For a system of differential equations, these points are found by setting each derivative equal to zero and solving the resulting algebraic equations. They represent population levels where predator and prey populations remain constant over time.

Predator-Prey Model Dynamics

Predator-prey models describe interactions between two species where one is the predator and the other the prey. The equations typically include terms representing natural growth or decay and interaction effects, such as predation. Understanding these dynamics helps interpret how populations influence each other and evolve over time.

Solving Systems of Nonlinear Equations

Finding equilibrium points often requires solving nonlinear algebraic equations simultaneously. Techniques include substitution or factoring to find all possible solutions. Mastery of these methods is essential to identify all steady states in systems like predator-prey models.